Closed-Constructible functions are Piece-Wise Closed

Abstract

A subset $B \subset Y$ is constructible if it is an element of the smallest family that contains all open sets and is stable under finite intersections and complements. A function $f : X \to Y$ is said to be piece-wise closed if $X$ can be written as a countable union of closed sets $Z_n$ such that $f$ is closed on every $Z_n.$ We prove that if a continuous function $f$ takes each closed set into a constructible subset of $Y$, then $f$ is piece-wise closed.